Abstract: We introduce a family of adaptive estimators on graphs, based on penalizing the L1 norm of discrete graph differences. This generalizes the idea of trend filtering [Kim et al. (2009), Tibshirani (2014)], used for univariate nonparametric regression, to graphs. Analogous to the univariate case, graph trend filtering exhibits a level of local adaptivity unmatched by the usual L2 based graph smoothers. It is also defined by a convex minimization problem that is readily solved (e.g., by fast ADMM or Newton algorithms). We demonstrate the merits of graph trend filtering through examples and theory.